Fourier Epicycles
circles on circles drawing a curve that never repeats

What you see on every new tab

Fourier Epicycles puts a chain of 16 thin grey circles in the middle of the New Tab page. The first and largest is centred on the screen, each next circle rides on the rim of the one before, and a bright pen dot at the very end draws a looping, glowing line on a near-black background.

Each circle spins one step faster than the circle it rides on, and neighbours spin in opposite directions, so the pen traces petals, knots and spirals that slowly reshape as the circles change size.

• Sixteen circles: From a large first circle, with a radius of up to a quarter of the screen height, down to tiny fast ones at the tip, each with a straight spoke joining its centre to the next.
• A 22-second trail: The pen's line stays on screen for about 22 seconds, thinning and fading as it ages, with a soft halo underneath.
• Slow rainbow: The line's colour shifts steadily through the hues, going all the way round roughly once a minute, so old and new loops show different colours.
• Breathing sizes: With no music, every circle swings between about half and full size at its own pace, which keeps the curve from settling into one fixed shape.

Drawn with the browser's 2D canvas at up to 1600 pixels wide, then copied to the screen through WebGL each frame. The chain is worked out six times per frame, so even the fastest circles draw smooth curves.

What it reacts to

• Mouse: Moving right speeds the whole chain up, from about one turn of the big circle every six seconds to about one every two seconds; moving up or down twists the circles' starting angles so the curve leans. Both ease in gently.
• Music: With music sync on, circles 1 to 5 swell with the bass, 6 to 11 with the mids and 12 to 16 with the treble, up to nearly double size. A kick makes the pen dot flare and draws a brighter stretch of line.
• Night: In Auto mode an indigo veil fades in around sunset and lifts around sunrise; the Day/Night button in the Themes panel can force either.

Free or Pro?Pro

Ambient New Tab is free to install, with five free themes. Fourier Epicycles is a Pro theme. Pro is $5.99 a month or $39.99 a year, cancel anytime, or $59 once for Pro Forever, and it unlocks every Pro theme, music sync, and every new theme as we release it. You can preview Fourier Epicycles full screen before you buy: open the Themes panel and rest your mouse on it. Changed your mind? 14-day refund, no questions.

How to get it

1. Add Ambient New Tab to Chrome.
2. Open a new tab and click the Themes pill in the bottom-right corner.
3. Pick Fourier Epicycles. Done.

Questions

What is an epicycle?

A circle whose centre travels round the rim of another circle. Stack enough of them with the right sizes and speeds and the end point can trace any closed shape; that is the idea behind a Fourier series. Here the sizes keep changing, so the shape keeps changing too.

Is Fourier Epicycles free?

It is a Pro theme. Pro is $5.99 a month or $39.99 a year, cancel anytime, or $59 once for Pro Forever. It unlocks every Pro theme, music sync, and every new theme as it is released.

Is the drawing really my song?

Partly. The music arrives as three levels, bass, mids and treble, and each level sets the size of one group of circles. Loud bass swells the big circles and makes wide sweeping loops; bright treble swells the small fast ones and adds ripples and tight curls along the line.

Why does the line disappear?

Each point of the line fades out about 22 seconds after the pen draws it, and the trail is capped at 2,400 points. That keeps the screen from filling up with old loops and keeps the drawing cost the same whether the tab has been open for a minute or a day.

Does it work on my phone?

No. It's a desktop browser extension for Chrome on Windows, macOS, Linux and ChromeOS. If you're reading this on a phone, send yourself the link.

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